NARRATOR: In earlier workshops
we've seen students successfully building on previous work
in mathematics to tackle problems of ever-increasing complexity.
In real-life classrooms it takes a long time to help students
build up this solid foundation of fundamental mathematical
ideas. Yet, no matter at which point the students arrive,
it is possible to begin the process. In many cases, it starts
with just one lesson. Let's look at some of these first activities.
Later in the program we'll see what the possibilities might
be.

[MUSIC]

MICHELLE DOUGHERTY: Oh I like
the way Jallell is showing me he's in listening position.
So is Adrian,..

NARRATOR: At the Quarles School
in Englewood, New Jersey, kindergarten teacher Michelle Dougherty
participated in a professional development workshop with Arthur
Powell, a researcher in the Rutgers long-term study.

MICHELLE DOUGHERTY: What we're
going to talk about now are the rods that you were using.
I'm going to see what you learned about the rods as you were
playing the game.

NARRATOR: Today, she is trying
out an activity that she has never done before, one that deals
with the concept of addition. Called trains, she asks the
students to find all the different ways of linking up shorter
rods so that their combined length will match the length of
a given longer rod.

MICHELLE DOUGHERTY: Who can
raise a quiet hand and tell me one way that we could make
the same size as the green rod using different colors. Adrian?

ADRIAN: Put a light green and
another light green on top of the light green.

MICHELLE DOUGHERTY: How many
light greens is that?

ADRIAN: Two.

MICHELLE DOUGHERTY: Can you
come up and show me how you would do that? I'll give you your
two light greens, and can you stand them on top of each other.
But before we do that, thumbs up if you think Adrian is correct,
thumbs down-- I don't think that's the same size. Jacob doesn't
think so.

JACOB: I do.

MICHELLE DOUGHERTY: Most people
do. Now you do. Oh, most people do. Let's see if Adrian's
right. I'm going to ask you to put it right here, stand them
on top of each other, and we're going to see if Adrian was
correct. Is he correct.

STUDENTS: Yeah.

MICHELLE DOUGHERTY: All right,
give him a hand. Nice job, Adrian. [applause] Nice job. And
he knew it in his head, and then he came up and he could show
us and he was right. Who knows another way we could use different
colors and it will be the same size? Jacob?

JACOB: Red-- I mean, one red
and two whites.

MICHELLE DOUGHERTY: One red,
two whites. Thumbs up if he's correct. You don't think so,
Tatiana ? Is one red and two whites going to be too small
or too big?

TATIANA: Too small.

MICHELLE DOUGHERTY: It's going
to be too small; what do you think? Too small or too big?

MICHELLE DOUGHERTY: They surprise
me all the time. I can't believe they came up with that many
ways to make the dark green rod. I didn't know if anyone would
have any ideas, I didn't know if they would want to try it,
you know. They're constantly surprising me with what they
know.

MICHELLE DOUGHERTY: Jacob,
talk to me. Stand up so everyone can see what happened.

STUDENTS: Too small. Too small.

MICHELLE DOUGHERTY: Oh no,
Thalia was right. It was too small. Jacob, what do we need?
What do we need to make it correct? [Suggestions from students.]

JACOB: One red.

MICHELLE DOUGHERTY: What Jacob?

JACOB: One red.

MICHELLE DOUGHERTY: Thumbs
up if-- is he right with one red? Thumbs down, is he wrong.
Emani you don't think he needs one red?

EMANI: No.

MICHELLE DOUGHERTY: What does
he need?

EMANI: Two.

MICHELLE DOUGHERTY: He needs
two reds? OK, let's see.

STUDENT: No, one, two white.

MICHELLE DOUGHERTY: You think
he needs two white?

JACOB: No, I use a red.

MICHELLE DOUGHERTY: Jacob wants
a red, so that's what I'm going to give him, and let's see
if he's correct. Go ahead. [Jacob laughs.] Is he correct?
Is that correct? Give Jacob hand. [Applause]

MICHELLE DOUGHERTY: If I continue
the discussion later, I'll see how many we can get and then
we can discuss how do we know when we're done-- like we discussed
over the summer in the math workshop, how do we know. And
I want to see what they say to that question. That will be
a whole other lesson: How do we know when we're done?

ARTHUR POWELL: In terms of
teaching mathematics and in terms of mathematics itself, it's
very important to get students involved in looking for justifications,
because at the heart of mathematics is the idea that we can
look at patterns and relationships and try to understand the
underlying reasons why those patterns and relationships exist.
And in reasoning, in understanding why they exist one is developing
ideas of proof.

MICHELLE DOUGHERTY: So how
many reds do you need?

JACOB: Two reds and two whites.

MICHELLE DOUGHERTY: Thank you.
Have a seat, Jacob. Jacob did it with two reds and--

STUDENTS: Two whites.

MICHELLE DOUGHERTY: I can't
believe how many ways we found so far. Boys and girls, we're
going to come back to this a little later-- because I know
you've been sitting for a long time-- to see if we can find
any more ways, or maybe- maybe we have them all. Raise your
hand in you think we have them all? We have them all. Hands
down. Raise your hand if you think maybe there's one more.
A couple people think there's one more. We're going to try
it later; I'll leave these here and we'll come back to it
a little later because you've been sitting for awhile.

MICHELLE DOUGHERTY: When I
experience the lesson first hand with the children, I seem
to learn more about the children and more about how I can
teach them and learn from them, because they're giving me
ideas. When they say, "I don't have any more of these rods,
I need more," they're giving me ideas. I need to figure out
a way that they can make that size rod. So, they're constantly
giving me more ideas on what I could use in my teaching.

[Music]

NARRATOR: Across town, at the
Lincoln School in Englewood, several other teachers have also
been expanding their classroom practice in response to their
work in Arthur Power's summer workshop. Melissa Sharp is a
first-year teacher, who is teaching a bilingual class of second
graders.

MELISSA SHARP: What are-- you
can touch the rods and figure it out. These are your rods--
all the different ways that we can make a train equal to the
length of one magenta rod.

NARRATOR: In this activity,
Melissa's students are also doing trains. How many trains
of shorter rods can the students make that will link up to
match the length of a given longer rod?

MELISSA SHARP: I was hired
two weeks before the workshop started and they said, "Do you
want to take a math workshop?" And I said, "Sure, I'm new;
why not?" It made me think more about the importance of clarity
in discussing things with students, instead of just letting
them say any vague answer, really being very specific, to
make sure that everybody knows exactly what that child is
talking about and really having the students think about the
problems they're working on, and not just simple problems
that they could figure out easily, but really have them working
hands-on with whatever it is that they're doing.

MELISSA SHARP: I see some of
you have already started writing down when you got your paper;
that's exactly what I'd like you to do, so--

NARRATOR: To help later full
group discussions, Melissa and her students are using letters
to keep track of their combinations. Arthur Powell continues
to assist Melissa and the other teachers with math activities
during the school year.

RITA: You can make another
one with the red and two whites. You already have that one
there. You already have that one.

MELISSA SHARP: Okay, Rita,
do you think you have all the ways?

RITA: Yes, because there isn't
any more smaller cubes except the lime, the red, and the whites,
and I already used all the ways.

MELISSA SHARP: How do you know
you used them in all the ways?

RITA: Because I tried all the
different ways.

MELISSA SHARP: You tried all
the different ways? Okay. So, could you use whites in any
other order? I see here you have all whites . Is there any
other way you would make a new train with whites?

RITA: No.

MELISSA SHARP: Is there other
place that you could put the lime rods with another color
rod to make one that equals the magenta rod?

STUDENT: No.

ARTHUR POWELL: I see Melissa
as being exceptional in one very important way: This is her
first year of teaching, the first time that she has responsibility
for a class, and Melissa seems to naturally want to investigate
how her students are thinking about problems. So, the questions
that she asked today in class drew out from each of her students
their statements as to how they were thinking about a particular
problem. She was always asking her students to justify their
statements.

STUDENT: Yes.

MELISSA SHARP: Why?

ARTHUR POWELL: She would ask
them, "Why do you think you have them all? How do you know?"
And those are questions which force students to reflect on
the work that they're doing and to develop an inner criteria
for knowing when they're right or wrong.

ALEXANDRA: To make a magenta
with red is two red, one red with two whites, and two whites
with one red, two whites and one red, so that's all the ways
you could do it with red.

MELISSA SHARP: How do you know
that's all the ways that you could do it with red?

ALEXANDRA: Because you put
another one of these it's going to be higher, and if you take
these it will be smaller. This is the only one to make this
one. And this one. You put this one in the middle like this,
it's going to be the same. It's the same thing but I have
it.

MELISSA SHARP: Right, you already
have it.

ALEXANDRA: So I need to change
it.

MELISSA SHARP: It's backwards.
Okay, great. I really like the way you thought about that
problem. She put a lot of thought into that, didn't she. So,
from that explanation, are you guys convinced that she has
found all the ways?

STUDENTS: Yes.

MELISSA SHARP: Okay. Excellent.

ARTHUR POWELL: What are they
using when they work on these problems? They're using their
visual imagery, they're using their ability to talk about
what they're doing, they're performing some actions; her students
are going to have a very rich background in understanding
of the additive structure of numbers. So, using facilities
that they already have-- they walk into the classroom with,
they then can engage in some very rich mathematical work.

STUDENT: I need to measure
the yellow.

MELISSA SHARP: I want them
to have the skills to figure things out on their own and not
just give up. I want them to be curious about the problems
and not just give any answer just for the sake of giving an
answer, and I want them to have the basic skills to be able
to do the things that they're going to need to do in the third
grade and the fourth grade and so on. I think that's most
important to me.

[MUSIC]

NARRATOR: Arthur continues
his work with fourth grade teacher Blanche Young, whose students
spent three sessions with the pizza problem. Two months later,
Blanche is introducing a new, but mathematically related,
problem, towers.

BLANCHE YOUNG: I'm going to
be giving out some connect snap-cubes. One requirement about
these snap-cubes. I borrowed them from almost every class
in the building, so when I give you connect-cubes or snap-cubes,
try not to mix them from one table to another. The assignment
is going to ask you-- here's your question- to make what is
called a tower. May I have one of the boxes from the top,
okay? And first we need to understand the terminology. This
is a tower, even though it's made with one unit, it's called
a tower--

NARRATOR: Blanche asks her
students to make as many different towers as they can by selecting
from blocks of two colors.

BLANCHE YOUNG: Your group is
going to be assigned to make as many four-tall towers as you
can, but when you're done, you have to agree that you have
gotten as many different arrangements as you can without exceeding
the height of four connect-cubes. Is that clear?

[simultaneous conversations]

BLANCHE YOUNG: Whatever color
that group is working on, you may give them these.

CHRISTINA: I'm making patterns.....
See look this is a Christmas color.

RAISA: I already did all of
those. See these two. Hold up- I messed up.

BLANCHE YOUNG: Do you recall
what the assignment is? To make how tall?

S: [simultaneous conversations]

BLANCHE YOUNG: Four tall, okay,
you're going to make them four tall.

RAISA: Ms. Young, I'm done.

BLANCHE YOUNG: How many did
you make?

RAISA: I made six.

BLANCHE YOUNG: Can you make
that same configuration in a different way?

RAISA: I did green, red, red,
green, and I need red. Two greens, and I need one more red.

BLANCHE YOUNG: The class is
a multi-ability group class. One of the positive is that there's
less stress on a particular skill. Students who may not be
able to recall their addition multiplication facts are able
to figure out a problem relatively easier than if they were
confronted with paper and pencil and you had to solve these
problems. So, if they have the opportunity to work with students
of various abilities, they draw from each other. They kind
of complement each other. And you could easily walk into a
classroom and not be able to figure out which students are
crackerjack math students and which students aren't. They
kind of level the playing field, I guess is a way to put it.

BLANCHE YOUNG: ...Four, 5,
6, 7, 8, 9, 10, 11 ,12, 13, 14, 15. You got to show me. Every
time you pick one and pair it off-- if I just picked these
two, why did I select these two?

TERELL: Because the one is
all like blue, and this one is all with white.

BLANCHE YOUNG: Yes. And then
you started selecting the others. What are you doing to select
the others?

TERELL: This one is one blue,
this one has one white at the top, this one has one white
in the middle, this one--

BLANCHE YOUNG: Good. All right,
can you finish then? All right, let's put these back. You're
going to finish-- Now, I'm having a little trouble with this
one because--

TERELL: This one is white.

BLANCHE YOUNG: Okay, at the
top there is--

TERELL: One blue, one white,
and two middle- two whites, two blues and it's with one white-

BLANCHE YOUNG: Very good. I
understand.

ROBERT: This one has three
blues on top and one yellow on the bottom. This one has three
yellows on the top and one blue on the bottom, and this one
has one blue on top and three yellows on the bottom.

ARTHUR POWELL: And those three
are different from any of the other towers that you've made?

ROBERT: These can be, like,
the same. These can be doubled, because these are together.

ARTHUR POWELL: Okay.

ROBERT: But they kind of go.
Hold up. One of these can go-no-no but this can go with this.
No, it can't.

ARTHUR POWELL: What do you
mean "can go with"? What do you mean by that?

ROBERT: They can be together.

ARTHUR POWELL: Why do you say
they can be together?

ROBERT: Like, this can be put--
I've got two of the same ones. Yeah, this one can be there
and this one can be here, or this one can be here, or there.
It can be in two places.

ARTHUR POWELL: Combinatorics
can focus their attention on using their skills of counting
to develop more efficient procedures. There are always different
ways of performing the counting procedures that they come
up with, so it provides a very ample area for students to
talk about the mathematics, to exchange ideas, and to display
their mathematical thinking.

ROBERT: --on top and one blue
on the bottom and one yellow in the middle.

CHRISTINA: Red, green, red,
red.

RAISA: I'm just going to do
the ones that are green.

BLANCHE YOUNG: What is it that
we're trying to change about the way the children-- they may
not even see this as math; you and I see it as math.

ARTHUR POWELL: -. Getting students
to take ownership of a problem-- you can see here that they
all own this problem. They're engaged in it very deeply, and
they're thinking deeply about the problem. Getting them to
see that they can exchange ideas among themselves and to listen
carefully to each other-- and we see some of that taking place.

BLANCHE YOUNG: I do need that;
I need that assistance with them taking ownership to a problem-solving
type math problem because they're quick to say, "I don't know
what to do. I don't understand that." And to make it meaningful,
to bring it down to their level so that they can own the problem,
get engaged in the problem, and work through the problem is
crucial to, crucial to right now, what's needed.

[MUSIC]

NARRATOR: At the Conover Road
Elementary School in Colt's Neck, New Jersey, fourth-grade
teacher Amy Martino was a researcher in the Rutgers long-term
study.

DREW: This equaled forty.

AMY MARTINO: What I really
pulled from that experience with Rutgers was, basically, children
know a lot more than we initially think that they do, and
if you can tailor learning to right where the child is at
in terms of their knowledge level, it makes all the difference
in the world. It's very important to listen to children and
this is something that we did for years. And it got to the
point where I just said, "You know, I really want to be doing
this full time, working with children."

AMY MARTINO: Good morning.

NARRATOR: On a typical morning,
as the students arrive, Amy greets them with a warm-up question.
Today's challenge is a math problem: Create as many different
equations as you can in which one side of the equation is
ten.

AMY MARTINO: This activity
that I call "Equations Galore" is an activity that I like
to do with the children. Today we picked the number ten; generate
all the equations you could that would equal ten. Initially,
this particular activity with equations was really meant to
be about a ten or fifteen-minute warm-up activity. It really
gives me a chance to sort out what needs to be sorted out
at the beginning of the school day and it gets them thinking,
it gets them going right away, as soon as they come through
the door.

NARRATOR: Surprised by the
wide variety of different responses her students have written,
Amy decided to re-arrange her schedule to allow time for a
class discussion.

AMY MARTINO: Boys and Girls-You
know what? Wherever you are at this point in terms of writing
equations, I'd like you to stop. Okay. This really was meant
to be a warm-up to get you going, and it certainly did. What
I'm going to ask is that we get some folks up here who maybe
want to share one of their favorite equations that comes out
to ten--

AMY MARTINO: Oh my goodness.
Look at that one. You want to read that one, Brian?

AMY MARTINO: Look at everyone.
Does that work? It's really, really something. I have a question
for the class before we move on. Take a look at Brian's; do
you think it makes a difference whether we read his from,
let's say, going from left to right, or say we started at
the end and went from right to left. Okay. Do you think we'd
come up with that same ten, either way?

STUDENT: Yes.

AMY MARTINO: Yes, no? Yeah,
Alexandra.

ALEXANDRA: Yeah, because when
you put all the numbers together you don't have to put them
in a certain order to get the same number- you just get the
same number. If you're doing five plus five-- I mean, like
five plus two, then if you-- that's seven, but if you do two
plus five it's the same thing as-- five plus two and two plus
five are the same thing.

AMY MARTINO: Okay, ... Tasha?

TASHA: I think it's sort of-
it would come out because if you have, like, seven plus two
plus three and then you did it three plus two plus seven,
it still comes out to the same answer, except it's in a different
order.

AMY MARTINO: Okay. Drew?

DREW: Well, she's saying by
pluses, but what about the minuses?

AMY MARTINO: Ah. Want to say
something more about that? What do you mean, what about the
minuses?

DREW: Because you're going
to have to have more numbers to minus then five thousand from
ten thousand?

AMY MARTINO: Okay, so you're
not a hundred percent sure that this is going to work going
both directions.

ALEX: It doesn't work.

AMY MARTINO: Alex, do you want
to say something else?

ALEX: I did it. It comes out
to twelve thousand and ten and that's not ten.

AMY MARTINO: All right, so
we ran into a problem there. Does anybody think that they
can see why-- in other words, what Alexandra said and what
Tasha said made perfect sense to me. You know, when you take
something like three plus five and five plus three, you do
get the same thing. What happened here?

AMY MARTINO Even though they
haven't learned about order of operations or any of the formal
ways of representing these more complex equations, they clearly
have some very, very elegant ways of thinking about equations
and how it all works.

AMY MARTINO: Tasha?

TASHA: It will work with pluses,
but I don't think it would work-- like, a long equation like
that, because you have pluses and minuses, and when I did
it backwards it equaled three thousand.

AMY MARTINO: Okay, so you're
saying then that it really does make a difference which direction
we start in.

AMY MARTINO: It is a lot more
work to do things this way. I believe you get a lot more than
the time that you're sacrificing for it; just the warm-up
activity that children were doing while I was basically taking
attendance and doing morning things, the kinds of discussion
and discourse that came out of that, they generated all kinds
of wonderful equations that equaled ten, we saw repeated subtraction,
we saw mixture of operations, we saw children generating patterns
that really showed that there were infinitely many possibilities
using certain operations. I mean, all that-- the power of
all of that as opposed to if I had given them, say, a math
worksheet to work on--

AMY MARTINO: Yes, Patrick.

PATRICK: I did it with pluses
but you have to stop a certain place, like it goes 4 plus
6, 5 plus 5, 8 plus 2, but then- 9 plus 1, but then you stop
there, you can't go really any farther. You have to go into
minuses.

AMY MARTINO: Okay. Paul do
you want to say something?

PAUL: Well you can go on with
addition but you'd have to go into the negative numbers, so
probably the next one would be negative one plus one- that
would be ten, eleven that is- would equal ten. And you'd keep
going on and on into the negatives.

AMY MARTINO: Okay. So, if we
decided to go into the negatives, Paul's saying you could
probably come up with more, do you agree with that Patrick?

PATRICK: Yeah.

AMY MARTINO : Yeah. Okay. Kim?

KIM: I had a pattern from division,
and I started with sixty divided by six, and I got the pattern
by, like, the first number of sixty is six, so I divided it
by six, and then I went up to seventy, divided by seven, and
I worked my way up to three hundred and ten.

AMY MARTINO: That's pretty
impressive.

KIM: And when I got three hundred
and ten, I divided it by thirty-one because they're the two
first numbers and it equaled ten.

AMY MARTINO: That equaled ten
as well. Very nice. Let me kind of pull you together for a
minute, okay?

[MUSIC]

NARRATOR: What actions, taken
by these teachers across the grade levels, seem to encourage
students to think mathematically? In what ways are these effective?

[MUSIC]

AMY MARTINO: -- there was some
question about it, and then the other side of the room did
everything in reverse and tried to see what was happening.

Part 2. "PASCAL'S TRIANGLE AND HIGH SCHOOL ALGEBRA"

STEPHANIE: Let's make a deal,
everything we make, we have to check.

DANA: I'll always make it,
and you'll always check it.

STEPHANIE: You make it and
I'll check it.

NARRATOR: How can students
make connections between the counting problems they do in
elementary school and high school algebra? One key is Pascal's
triangle, which is often included in the traditional high
school algebra curriculum.

STUDENT 1: ... One, 6, 15,
20, 15, 6, 1.

STUDENT 2: All right, now I
see it.

NARRATOR: When the Kenilworth
Focus Group students first approached the combinatorics problems,
they often began by looking for patterns.

BRIAN: Well, once when we find
one, we just do the opposite of it.

ALICE ALSTON: What do you mean,
"the opposite?"

NARRATOR: Some of the patterns
that emerged were finding opposites, pairing each tower with
an identical tower with opposite colors.

BRIAN: When found this one
out, we just put two blues on the top and three whites in
the middle.

ALICE ALSTON: Oh. Do they always
have an opposite?

ROMINA: Yes.

BRIAN: Yeah.

NARRATOR: Another pattern is
grouping, grouping towers in subsets, defined by the number
of blocks of a chosen color.

BRANDON:... Three's group,
and then switch those around, same thing.

NARRATOR: In the Kenilworth
study, the students went on to find a further pattern, that
each time the number of possible blocks increases by one,
the number of possible towers doubles.

STEPHANIE: Well it goes like
in a pattern, you have the 2 times 2 equals the 4; the 4 times
2 equals the 8, and the 8 times 2 equals the 16.

CAROLYN MAHER: I wonder why?
If this is a pattern, what would you guess would be with Towers
of 5?

STEPHANIE: If I had a guess?

CAROLYN MAHER: By noting this
pattern.

STEPHANIE: ... Thirty-... yeah,
thirty two.

CAROLYN MAHER: You would guess
32.

STEPHANIE: I would guess 32.

[Music]

NARRATOR: All of these patterns
are represented in the array of numbers called Pascal's triangle.
Blaise Pascal, a French mathematician, wrote about this symmetric
triangle pattern in 1654, in the course of investigating probability
problems.

To build Pascal's triangle,
we start with 1. The first row of Pascal's triangle has two
1's. The second row starts with a 1 at each end, but also
includes the number that is the sum of the pair of numbers
directly above. We continue this pattern of starting with
a 1 at each end, and adding every pair of numbers from the
previous row to make every number in the new row.

[Music]

NARRATOR: Pascal was not the
first to notice this array of numbers. It was discovered in
China as early as 1150, A.D. and was documented before Pascal
in Japan, Persia, and Germany.

At the Ferris High School
in Jersey City, New Jersey, a math teacher uses Pascal's triangle,
along with combinations activities, to help her students grasp
difficult concepts in algebra. Gina was a researcher in the
Rutgers long term study.

GINA KICZEK: ..the different
combinations? OK.

GINA KICZEK: My name is Regina
Kiczek. I've been teaching since 1972. I've been in Jersey
City since 1980, teaching here at Ferris High School. I've
changed as a teacher over time. The more I have grown accustomed
to listening to what students have to say and asking questions
that will elicit their thinking, the more I've come to understand
that the way I see something is very rarely the way that they
see it.

STUDENT: It's like a pattern.

GINA KICZEK: It's like a pattern?
Okay. You discovered something about this pattern?

NARRATOR: Gina's algebra II
class is mid-way through a unit that includes Pascal's triangle
and the binomial theorem. The binomial theorem describes how
to multiply or expand an expression such as a + b to any power.
These expansions are frequently used in algebra, but they
quickly become tedious if one has to multiply them out by
hand.

Pascal's triangle can provide
a shortcut for finding the co-efficients of the terms in binomial
expansions. However, Gina introduces Pascal's triangle only
after the students have experienced a number of counting activities
such as towers and pizzas.

GINA KICZEK: Okay, tell me
about towers. How does 2 to the n work with towers?

STUDENT 1: There's two different
colors.

GINA KICZEK: It's two different
colors. And, what's the n?

STUDENT 2: It could be like
5 high.

GINA KICZEK: Okay. So if it's
5 high, and it's 2 different colors then it's 2 to the ?...

STUDENT 2: Fifth Power.

GINA KICZEK: Fifth Power?...

GINA KICZEK: In the standard
algebra II curriculum that we follow, the binomial theorem
is basically the end of the course. And it's something that
we touch upon, and basically, the students just expand a +
b to the second, the third, the fourth, the fifth, look for
a pattern. Some students will have seen Pascal's triangle
before, some students will have not. It depends on the class.
But basically, the idea is get to the binomial theorem, talk
about expanding binomials, and then you're done, that's the
end of the year, let's review.

I think that the problems
that we do-- towers, pizzas, ice cream-- I think that that
helps to give them an intuitive understanding. And I think
that the concrete objects and the other types of things that
they've been working on, I think that gives them a really
good basis for understanding why it is that this happens.

GINA KICZEK: All right. So
what I'd like you to do is go to your tables, to your groups...

NARRATOR: On this day, Gina
is giving her students a new challenge. If ice cream is served
in bowls that can hold up to 6 scoops, how many different
ways can the ice cream be served? The students start by assuming
that they have no more than 1 scoop of each flavor.

The groups of students apply
their previous knowledge of Pascal's triangle toward this
new problem.

GINA KICZEK: So let's see,
you're telling me that this 64 different choices for bowls
of ice cream? And you don't have to write anything but numbers?

STUDENT: Uh-huh.

GINA KICZEK: Okay. So this
is kind of valuable if it works. Can you tell me what each
of these numbers stands for, in terms of bowls of ice cream?

NARRATOR: The students' previous
work with pizzas and towers has led them to a central idea
of Pascal's triangle, [Music] that the numbers within each
row map to the subsets of possible combinations.

For example, let's look
at the fourth row of Pascal's triangle. When building towers
4 high, there are 4 towers with 1 blue block, 6 towers with
2 blue blocks, and 4 towers with 3 blue blocks. Likewise,
every number in every row of Pascal's triangle corresponds
to the number of possibilities in a grouping of towers. Added
together, the sum of the numbers in each row equals the total
number of combinations possible.

GINA KICZEK: ...so that's no
ice cream, what's the six?

STUDENT 1: One kind of ice
cream. One flavor.

GINA KICZEK: Okay.

STUDENTS 1: Then 2, 3, 4, 5--

STUDENT 1: There's 6 -all flavors.

GINA KICZEK: So this last one
stands for all six?

STUDENT 1: Mm-hhmm. All of
them.

GINA KICZEK: So how many do
you have altogether, then?

STUDENTS: Sixty-three.

STUDENT 2: No, because you
don't count that one if that one doesn't have any, right?

STUDENT 3: 63.

NARRATOR: By not counting the
empty bowl, the students come up with 63 possibilities. In
the second half of the 80 minute block, Gina presents a new
problem, where the order of flavors does make a difference.

STUDENT 1: The cones were delivered
later in the week. The owner soon discovered that most people
who order cones are particular about the order in which the
scoops are stacked.

NARRATOR: The problem is, how
many different ice cream cones can be made, using up to 4
scoops of ice cream, by selecting from 6 different flavors?

STUDENT 2: So we had to use,
like-- we've got to use, like, these flavors and, like, we
use 4 instead of 6? Right?

STUDENT 1: Yeah.

STUDENT 1: Ms. Kiczek it says
for this one right here that they like- like if it's vanilla
over chocolate, or chocolate over vanilla. But I don't understand
it, because we're just supposed to double it or something?

GINA KICZEK: Well, I don't
know. What do you think? If you had vanilla and chocolate,
chocolate and vanilla, is that two different things?

STUDENT 1: It's the same thing,
but you're just switching it around.

GINA KICZEK: Okay. In a bowl,
does it matter?

STUDENT 1: No but in a cone-

GINA KICZEK: No, but in a cone,
does it matter? To some of these people it does.

STUDENT 1: Yeah.

GINA KICZEK: To some of these
people it does, okay? Let's think about three scoops now.

STUDENT 2: You can start with
this one on top, and then like that-- the chocolate, the cherry,
and this one. That's one cone.

GINA KICZEK: Okay.

STUDENT 2: Then you can stay
with that one, but instead this one, reversed that?

GINA KICZEK: Okay.

NARRATOR: By looking at the
number of possibilities or permutations with 3 scoops of 3
flavors, these students are making a solid first step to solving
the problem. Extending the work to 6 flavors and including
all the other possibilities-- 1 scoop, 2 scoops, and so on--
will eventually lead to the solution.

STUDENT 2: So it's kind of
like when we did the towers, right? Like if you have 4 high
and 4 yellow, and then you put 1 blue-- so you keep moving
it down.

GINA KICZEK: That's a possibility.

GINA KICZEK: As we moved through
the combinatorics unit, every problem builds on what they've
done before, but there's always a new wrinkle in it. What
happens, for example, today, with the cups and the cones,
will-- what about this new wrinkle now with the cone-- let
them think about something new, let them build on what they've
already done, and then learn to express themselves, write
down something, and think about it.

GINA KICZEK: Okay. What do
you think about that?

STUDENT 2: Oh yeah there has
to be 3 more flavors.

GINA KICZEK: Oh, I see what
you're saying.

STUDENT 2: So you've got to
do more-- six more, so there's 12 altogether.

GINA KICZEK: I don't know,
you've got to decide that.

NARRATOR: How does an intuitive
understanding of problems like towers, pizzas and ice cream
help students with higher mathematical ideas?

STUDENT 2: No, you get more
because then you get that group with this group, and you mix
those.

STEPHANIE: All right. We went
back to the beginning with the towers, and we went way back
to when we were building towers, like, a long time ago.

NARRATOR: In March of 1996,
mathematician Robert Speiser, of Brigham Young University,
interviewed Stephanie. Stephanie linked towers 2 high to each
number in the second row of Pascal's triangle.

STEPHANIE: We figured out all
of them, like, from this.

ROBERT SPEISER: OK. Tell me
a little more about the triangle. Okay, does this have to
do with towers?

STEPHANIE: Yeah.

ROBERT SPEISER: Show me the--

STEPHANIE: It would be-

ROBERT SPEISER: So these are
the towers that are 2 high-- 2 blocks high. And then how do
you find the 1, the 2, and the 1?

STEPHANIE: It would be-- if
you're selecting green-- it would be 1-- Well, if you're selecting
blue, it would be 1 with no selections of blue, 2 with 1 selection
of blue, and 1 with 1 all selections of blue. It's like the
towers.

ROBERT SPEISER: It's like the
way you'd organized the towers before.

STEPHANIE: Mm-hmm. Yeah.

ROBERT SPEISER: How would you
organize the next row so that it makes more sense-- so that
it makes the most sense for you?

STEPHANIE: Oh, to the chart-...
it would be-... Wait.

ROBERT SPEISER: How did you
know to write those numbers?

STEPHANIE: Because 1 goes to
1 and 1, and then 1 goes here, 1 + 1 is 2, and 1 goes there.

NARRATOR: Stephanie then showed
how adding either a green block or a blue block can make towers
3 high and lead to a new row of Pascal's triangle.

ROBERT SPEISER: Did you explore
why the adding works?

STEPHANIE: Its choices can
be green, built onto it-- it can either have a green on top
of it or a blue on top of it. And there was no one with green,
blue, blue. That's why.

ROBERT SPEISER: Good. It looks
to me like the others worked the same way.

STEPHANIE: Yeah, you just keep
building on.

NARRATOR: This is called the
additional rule of Pascal's triangle. The same addition rule
applies to polynomial co-efficients.

[Music]

NARRATOR: In January of 11th
grade, the Focus Group of five Kenilworth students met after
school to work on a problem they had never seen before: the
World Series problem.

In the World Series, assuming
two teams are equally matched, and the first team that wins
four games wins the series, what is the probability that the
World Series will be won in four, in five, in six, and in
seven games?

ROMINA: Why don't we do like-
you know how we do like write out the blues-

CAROLYN MAHER: We'll leave
you alone.

JEFF: Yeah, that's what I'm
saying-

ROMINA: So that you can go
all 7, because if you go all 4, it's only A, A, A. A, A, A,
A, and B, B, B, B. Team A and Team B? Those are the only possibilities
for four.

GINA KICZEK: We had worked
with the students on a lot of different combinatorics problems,
in towers and pizza, and extensions of those things. So we
decided to see what was possible. Given all of the different
ideas that they had built, we wanted to see if they could
solve a particular probability problem without having been
taught how to do it, without any formal rules or notation
or anything being imposed. We just wanted to see what would
happen.

ROMINA: So in 4 games, it would
be like 1/2 of a chance? Or would we have to write out, with
using all 7?

JEFF: See, I think that it's
the hardest doing it in 4 games. Definitely hardest. So that
wouldn't be one half.

BRIAN: Wouldn't it be the odds
of winning 1 game, times odds of winning one game, times odds
of winning game, times odds of winning one game?

JEFF: That's what I'm thinking.

ANKUR: It's a 50 percent chance
of winning the first game.

BRIAN: All right. So it's like
a half times a half-

GINA KICZEK: They did the problem
in about an hour, and they did it correctly, and I've been
studying the tape for about two years. There's a lot of mathematics
on the tape. And I'm looking at not only what they did to
solve the problem, but I'm trying to look for the origins
of those ideas.

BRIAN: Just remember, the odds
get harder to win 2 in a row, like a coin flip.

ROMINA: Yeah, that's how you
do it. Half times half times half times half.

NARRATOR: Their answer, 1/16,
was added to another 1/16 to account for both teams.

ROMINA: Would we do that for
5 games? That would be-- Yeah, there's going to be a lot.

NARRATOR: Mike worked on his
own, using Pascal's triangle, while the other students worked
together.

ROMINA: Would it be, like,
say, the probability of something, and then it would be like
B, B, B, B. And any ones that have B, B, B, B--

JEFF: Yeah, then that would
be that number and that number. That's what I was thinking.

ANKUR: So we've got to do it
like that.

NARRATOR: Moving on to 5 games,
Romina proposed writing out all combinations, using strings
of A's and B's to represent the wins.

ROMINA: Yeah, I know. I'm just
saying, like, each time we look over, like, five, well, we'll
see how many. You know?

GINA KICZEK: Of course, for
a 4 game series, it's pretty easy. You either have 4 wins
in a row for this team or 4 wins in a row for that team. And
for a 5 game series, it was a little bit more complicated,
and they realized that they got 8 different strings. But when
they tried to figure out what the probability of that was,
they knew it was 8 over something, and it was the 8 over something
part that they had a little trouble with.

ANKUR: They have 8 ways of
winning, but it would be

over--

JEFF: Oh, 8 over 1-- No, how
do we find out?

ANKUR: Be over the total possibilities
of 2...- 2 colors and 5 things.

GINA KICZEK: They seemed to
have the idea that probability is the number of favorable
outcomes over the number of total outcomes, although they
never said that, they never had that definition. But it was
an intuitive type of thing that they seem to have been doing.

ANKUR: Know what I'm talking
about or no?

JEFF: Yeah, it's got to be
over 2. The total possibility's 4 spaces.

ANKUR: Yeah, 4 spaces.

JEFF: Yeah, all right, it makes
sense-- And that would be 8 over 2 to the fifth, do you think?

ANKUR: That's 16.

JEFF: And then 8 over 2 to
the fifth?

ANKUR: I guess.

JEFF: Which would be 32.

MIKE: Is there's 32 possibilities
for 5 games.

JEFF: Yeah. That sounds--

ANKUR: I think there's more.

BRIAN: For how many games?

JEFF: Five.

ROMINA: Hold on. You've got
8?

JEFF: 5 spaces.

ANKUR: Total possibilities.

JEFF: 32 for 5.

GINA KICZEK: So then they got
to a 6 game series. That was a little bit more difficult to
list all the different possibilities for 6 games, but they
did it. When they got to the 7 game series, they realized
that that was going to be a lot to count.

JEFF: You see doubles in that?
I can't even look at it.

ROMINA: You want me to read
them?

ANKUR: For 7?

ROMINA: With "A" winning.

ANKUR: Did you just randomly
write them, or did you do them in some order?

ROMINA: I started in some order,
then I-- It's hard though, because you're just, like-- I don't
know. Did you write them all out?

ANKUR: I wrote them out.

ROMINA: Oh, you did?

ANKUR: I wrote out 10.

NARRATOR: Ankur found out that
his winning probabilities for 4, 5, 6, and 7 games added up
to 1.

ANKUR: It is right. 40 out
of 128. The whole thing adds up to 1.

BRIAN: Do they match with them?

ANKUR: They match.

JEFF: Wait, 40 out of 128?

ANKUR: Yeah, it works.

GINA KICZEK: They looked at
it in cases-- 4 game, 5 game, 7 game series. They got the
probability of each one individually. They saw that they gave
them a total of 1. They knew that that was supposed to happen.
And they were ready to present their solution, all using representations
that were basically retrieved from earlier investigations,
and maybe modified a little bit to fit the situation.

JEFF: So basically, what we
did was, that could be 2 possibilities, that could be 2 possibilities,
that could be 2, that could be 2. And that was like where
we went back to the old days, and it was like 2 to the n.
So 2 X 2 X 2 X 2. That's how we got 16. And that would be
the bottom number. And then in order win the 4 games, these
have to be either all A's or all B's. So we got 2 out of 16,
for winning at 4 games, which is probability of winning in
4 games. That make sense?

MIKE: They have something that
works for that first one, but does it work for-

JEFF: Yeah. We're going to
go on. So for the next one, we're going to do the same situation,
but this would be 2 to the 5th. So that's going to be out
of 32. And 32's the bottom number. And then, I think for these
we were just kind of-- we went through them. That's why there
are strings of A's and B's on everyone's paper. In order to
get these, we went through all the possibilities where there
was 5, 5 places, and A or B was in 4 of them. And we went
through all of them, and that's how we got that. And then
we ended up with 8 of 32 put for that. Now that's not too
convincing, because we just went through them. But we went
through all the ones that were out of 5, with 4 A's. And that's
how we got that. I don't think we have a real, concrete mathematical
backing to that.

NARRATOR: At this moment, Mike
presented his approach. Mike used Pascal's triangle to explain
his strategy.

MIKE: I just found, like you
take the fourth number of each one. For some reason if you
double each number, because you have 2 teams, you get the
possibilities for 4 games, 4 games- equals two, right? You've
got 8, 20, and 40 like they said. Those last- those 3 games
that they won, the first 3 games, if they win that, that would
be like there's 3 possibilities- would be- if they win the
next game- or if they win- I don't know how to explain this.
On the third game...I don't know.

JEFF: I guess if we were going
to say-- if was out of 8 games, then there would be 35? The
probability would be 35 out of-- you know what I'm saying?

ANKUR: Yeah.

BRIAN: Yeah.

MIKE: It would be 1, 7--

ANKUR: Just add the 15 and
20 for 35.

JEFF: So I mean, there's got
to be something there, because it wouldn't all-

MIKE: It would be 35 doubled.

ANKUR: Yeah.

JEFF: Yeah. 35 for one team.

MIKE: But the limits of the
problem are you have to win 4 out of 7. Not 4 out of 8.

JEFF: Oh yeah, I know.

GINA KICZEK: So Michael notices
in the triangle that on one of the diagonals, he finds the
numbers 1, 4, 10, and 20. And the counts, in each case, the
count for a 4 game series, the number of ways you can win
a series in 4 games was 2, and the number of ways you could
win it in 5 games was 8, and in 6 games was 20, and in 7 games
was 40. So he's got 1, 4, 10, and 20 in this diagonal. And
if you double them, he said that that's 2, 8, 20, and 40.
"So there's obviously some connection," he said. "But I don't
know what it is yet."

So they spent some time
looking at that connection. I think initially it's just an
interesting insight on Michael's part. He notices a pattern
there. And noticing that connection spurred all kinds of activities
over the next three sessions. That allowed them to do some
pretty sophisticated mathematics.

NARRATOR: In May of their junior
year, Kenilworth High School students returned to school one
evening around 7:30 p.m. for a research session with Carolyn
Maher and her colleagues from Rutgers University.

Carolyn began the session
by asking them to review what they had discussed in their
pre-calculus class earlier that day.

The class had touched on binomial
expansions, and the students had learned about a way to calculate
the co-efficient of any term without having to write out Pascal's
triangles. The notation is called N choose R. It evaluates
how many ways there are of choosing R objects from a set of
N objects.

Mike drew Pascal's triangle,
and explained how the numbers could be assigned to the N choose
R notation.

MIKE: All right. This would
be like 3 choose 1. How many different places could you put
that 1, that one guy- there's only one place. The next one
would be 3 choose 3. There's obviously 3 different places-

CAROLYN MAHER: You 3 choose
what? What's the next one?

MIKE: -3 choose 1. The next
would be 3 choose 2, which you just figured that out-- is
3. And the last one is 3 choose 3. You can only put those
3 people in 3 places. You can't-- no other place to put them.

CAROLYN MAHER: I have another
question. You could write more rows of that triangle. And
now you're telling me you can write them as the "choose" way,
you've called that. So can you take, let's say, another row
or two? And show me the addition rule, and what it looks like,
with your new notation for a particular row.

MIKE: Add this and this, and
go like that?

CAROLYN MAHER: Sure. Or 3 and
3 is 6. Show me what that looks like with that new notation.

MIKE: All right. Let's go to
this one. This would be, like, 3 different places, I guess.

JEFF: Which one are we looking
at?

MIKE: That one right there.

JEFF: That would be a plus
b to the third?

MIKE: Let's say you have--
like, here's a number, right? Zero means no toppings, 1 would
be a topping. So the first category is everything with no
toppings, and that's your number for that one. This is like
binary numbers or something, Next would be- there's all the
ones that have 1 topping.

JEFF: Mike, you got to make
that a zero at the end. You messed up.

MIKE: What? I knew that. There's
your 3 choose 1, and there's 3 different combinations you
can put that. And I can go on forever doing this. But when
have a new- when you add another place, another topping-

JEFF: That can be one or the
other- one or the other- one or the other-

MIKE: So it could be one or
the other. It could be a zero or a one, a zero or a one, a
zero or a one. So all these 3's would either move up a step
onto the next category, and have 2 toppings, or they might
stay behind and still only have 1, if they have the zero.
So 3, I get a topping-- go to this one. And 3 won't- will
stay. And obviously, this guy's going to get a topping; that's
why you add this one. So now this guy's going to have-- without
toppings-- you're going to add a topping onto him-- and it's
going to be 1 topping. These 3 with 1 topping won't get one.
So, you know, you can put them in the same category as this
one, that's 4.

JEFF: Yeah. Those are 4.

MIKE: And, you know, the 3
that had 2 toppings won't get any.

JEFF: Yeah. So that'll go to
the left?

MIKE: And you'll put them together
with the ones that did get some. That's why you would add-
keep on adding.

CAROLYN MAHER: Well I want
you to show me how the addition rule works, in general.

JEFF: N choose X plus N choose
X + 1

MIKE: -Equals that

JEFF: -plus 1 equals that right
there. Well that's because this would

be gaining an X and going into
the X + 1, and this would be losing an X.

MIKE: No, no, no-

ANKUR: That stays the same.

JEFF: That's staying the same, and that's- is the X + 1

MIKE: And the toppings going
to change because you have more-

JEFF: -because you have more
things. And why do it? -Because when you add another topping
on to it, say the toppings were one and zero, if it gets a
topping, that's why it goes up to the X + 1, and since it
doesn't get anything, it will stay the same. And in this one,
it's staying the same, right? And that's why it's going there,
like saying that's the zero, and going to there. Make sense?

BRIAN: Yes, it actually does.

JEFF: So that would be the
general addition rule, in this case.

CAROLYN MAHER: In fact, I wish
someone would do it on the board on the right there, write
that addition statement, using factorial notations.

JEFF: Minus X plus- exactly.
You know like, how intimidating this equation must be, like
if you just pick up a book and look at that?

CAROLYN MAHER: Could you very
carefully check that arithmetic?

MIKE: You think we're wrong?

ANKUR: Yeah, it's right there.

JEFF: Where is it?

ANKUR: It's right above n over
x.

MIKE: There you go.

CAROLYN MAHER: You sure?

MIKE: Yeah, I'm sure. You got
anything else?

NARRATOR: How do the Pizza
problems, Towers problems, and World Series problem relate
to Pascal's Triangle?